# Egyptian Formula for Area of Quadrilateral

## Theorem

Let $\Box ABCD$ be a quadrilateral.

Let the sides of $\Box ABCD$ be $a$, $b$, $c$ and $d$ such that $a$ is opposite $c$ and $b$ is opposite $d$.

Then the area of $\Box ABCD$ can be approximated by:

$\map \Area {\Box ABCD} \approx \dfrac {a + c} 2 \times \dfrac {b + d} 2$

The closer $\Box ABCD$ is to a rectangle, the better the approximation.

## Also known as

This formula can also be seen as the Roman-Egyptian, Egyptian-Roman or Roman formula.

## Historical Note

This formula was developed as a pragmatic way to determine the area of fields which were not quite rectangular, but were a close enough approximation.

The Egyptians were capable enough surveyors to be able to make accurate right angles, and so marking out a field which is more or less a rectangle was not a difficult exercise.

But there are frequently inaccuracies, and so it is challenging to obtain an accurate measure of the exact length and width of such a field which may or may not be precisely rectangular.

However, the length of each side is a straightforward measurement.

As long as the angles do not deviate "too far" from being right angles, the formula is accurate enough to be of practical use.

The Romans adopted this formula, and it sometimes bears their name.